Optimal estimates for the gradient of harmonic functions in the multidimensional half-space

TL;DR

This paper derives a precise formula for the best possible bounds on the gradient of harmonic functions in a multidimensional half-space, based on boundary data in various L^p spaces, enhancing understanding of harmonic function behavior.

Contribution

It provides a new representation of the sharp constant in gradient estimates for harmonic functions with boundary values in L^p spaces, specifically for p=1, 2, and infinity.

Findings

Explicit formulas for sharp constants when p=1, 2, and infinity.

Improved understanding of gradient bounds in harmonic analysis.

General framework for boundary value estimates in multidimensional half-spaces.

Abstract

A representation of the sharp constant in a pointwise estimate of the gradient of a harmonic function in a multidimensional half-space is obtained under the assumption that function's boundary values belong to $L^p$. This representation is concretized for the cases $p=1, 2,$ and $\infty$.